An engel condition with skew derivations

Let R be a prime ring and set [x, y]₁ = [x, y] = xy - yx for x, y ∈ R and inductively [x, y]_k = [[x, y]_k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x]_k = 0 for all x ∈ L, where k is a fixed positive integer, then charR = 2 and for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.